Theoria motuum planetarum et cometarum. Continens methodum facilem ex aliquot observationibus orbitas cum planetarum tum cometarum determinandi. Una cum calculo, quo cometae, qui annis 1680 et 1681. Itemque ejus, qui nuper est visus, motus verus investigatur.

Berlin: Ambrosius Haude, 1744.

First edition of Euler's first treatise on astronomy, “a fundamental work on calculation of orbits” (DSB). Stimulated by the appearance of two great comets in 1742 and 1744 (now designated C/1742 C1 and C/1743 X1), Euler developed new methods to determine the (elliptic) orbits of planets and the (elliptic and parabolic) orbits of comets. His first major contribution in the present work was to the ‘two-body problem,’ the problem of determining the motion of two spherical bodies under their mutual gravitational attraction (such as the Sun and a planet). Newton had attacked the two-body problem using geometrical methods in Principia, and preliminary analytical results had been presented in 1734 by Daniel Bernoulli, but it was Euler in the present work who gave the first complete analytical solution. The second major contribution of the present work was the introduction of new techniques of perturbation theory – the method of successive approximations that Euler used to determine parabolic orbits is still known as ‘Euler’s method.’ Euler used these new techniques, together with observational data supplied by Alexis-Claude Clairaut in Paris, to calculate the orbits of the comets of 1742 and 1744, and his success stimulated others to use his methods to predict the next return of Halley’s comet, which Edmond Halley had first observed in 1682. “When Euler reported back on his successful calculation of an orbit from their data, the Parisian astronomers, even die-hard Cartesians like Jacques Cassini, had to accede to the power of Newtonian theory. In fact the French adopted it with such enthusiasm that they virtually took over the work on Halley’s comet at its forthcoming apparition, Clairaut foremost among them” (Broughton, p. 126). As mentioned in the title of the present work, Euler also applied his methods to check the orbit of the great comet of 1680/81 – Newton had initially believed that the observations of 1680 and 1681 were of two different comets, but eventually agreed with Flamsteed that they were of a single comet that initially approached, and then receded from the Sun. The treatment of the full three-body problem had to wait another decade, until the appearance of Euler’s first theory of the motion of the Moon, presented in his Theoria motus lunae (1753).

“The year 1744 saw the publication of Euler’s second Berlin book, Theoria motuum planetarum et cometarum (Theory of the motions of planets and comets), his first on astronomy. This smaller, 187-page treatise was printed anonymously after another comet was sighted on 18 January, the fourth sighting since 13 December …

“The Theoria motuum begins with the orbit of Mercury and makes a table of interpolations from the radius vector to compute eccentricities more closely. Euler calculated the interpolations with logarithms, trigonometric functions, and fourth-degree roots. He developed the first differential equations for computing every point in the orbits of Earth and Mars. Thereby he illustrated new methods for examining planetary perturbations. Using the latest telescopic observations, Euler differentiated comets from fixed stars and described the research of the time on both of them. While the mass of comets was still unknown, he found their orbits to be nearly parabolic ellipses and sought to devise functions for computing each course element, including anomalies. A much-elongated ellipse was assumed if comets were permanent. But Euler found their path closer to a parabola. Comparing accurate observations of the comet of 1742, constructing variations of similar ellipses and parabolas, and employing records concerning comets in 1743 and 1744, Euler revised and improved upon Jacques Cassini’s method for computing the time at which a comet reaches perihelion, its nearest point to the sun. After gathering the latest observations, Euler concentrated on formulating new differential equations that better traced the paths of comets. The application of his equations and formulas provided the most accurate computation of most points in the orbits of planets and comets to that time. But the Theoria motuum also cites unresolved problems in constructing a precise, theoretical account for comets’ entire orbits; Euler recommended caution toward the belief that comets foretell the wrath of God, and rejected the notion – even though they could come close enough to do so – that they would ever destroy Earth, citing the Bible as denying that this could happen.

“This work was formulated entirely as a two-body problem and not the classical three-body problem that had drawn Euler’s interest as early as 1730. Until the nineteenth century, the book was fundamental for calculating the orbits of planets and comets. It posed severe difficulties for the publisher Ambrose Haude in Berlin. A small but difficult work, it contains errors in the printing of computations and of some formulas; through footnotes, Euler attempted to correct many of these” (Calinger, p. 229).

Using the techniques developed in the present work, Euler went on to develop his theory of the Moon’s motion which Tobias Mayer used to constructed his important lunar tables. These were bequeathed to the English Board of Longitude on Mayer’s death in 1762; their great accuracy allowed longitude to be found within a few nautical miles and also permitted the position of the Moon to be calculated several years in advance. For this Mayer’s widow in 1770 received £5000 from the English parliament, and in recognition of Euler’s theoretical contributions a sum of £300 was also voted as an honorarium to him.

The remarkable engraved frontispiece, by Berol after F. H. Fritsch, depicts the solar system with the Sun as one among many other stars in a plurality of worlds.

Leonhard Euler (1707-83) was the most prolific mathematician of all time and one of the most influential. “He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos, etc. as functions rather than as chords as Ptolemy had done. He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz’s differential calculus and Newton’s method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three-body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics” (Mactutor). “Euler’s studies in astronomy embraced a great variety of problems: determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, considerations on the physical nature of comets, and the problem of retardation of planetary motions under the action of cosmic ether. His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time” (DSB). The great French mathematician Pierre-Simon Laplace advised to “read Euler, as he is a master of us all”.

Eneström 66; Honeyman 1063; Houzeau & Lancaster 11948. Broughton, ‘The first predicted return of Halley’s comet,’ Journal for the History of Astronomy 16 (1985), pp. 123-133. Calinger, Leonhard Euler. Mathematical Genius of the Enlightenment, 2016.



4to, pp. [3], 4-6, 9-187 (i.e., 188: last page mispaginated 187), with engraved frontispiece and four folding engraved plates of diagrams. Woodcut vignette on title, woodcut initials and head- and tail-pieces. In this, as in all copies we have seen, the frontispiece, which was printed on A4, has been cut out and bound facing the title. Pages 7 & 8 are therefore omitted, but the text is continuous and the volume is absolutely complete. Contemporary boards, with some light wear, a very good copy.

Item #2726

Price: $7,500.00

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